When \(\varphi (s)=s^{1-p}, k=n-1\) , Eq. (REF ) reduces to the \(L_p\) -Minkowski
problem, which has been extensively studied, see e.g. [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}.
